Linear Transformation from R3 to R3

[jolly_math]

Homework Statement

Describe explicitly a linear transformation from R3 into R3 which has as its
range the subspace spanned by (1, 0, -1) and (1, 2, 2).

Relevant Equations

linear transformation

"There is a linear transformation T from R3 to R3 such that T (1, 0, 0) = (1,0,−1), T(0,1,0) = (1,0,−1) and T(0,0,1) = (1,2,2)" - why is this the case?

Thank you.

 

[Hall]

Does it help to glance at the following matrices:
$$
\begin{bmatrix}
1 & 1 & 1 \\
0 & 0 & 2 \\
-1 & -1 & 2\\
\end{bmatrix}
\times
\begin{bmatrix}
x \\
y \\
z\\
\end{bmatrix}
$$
?

 

[PeroK]

Homework Statement:: Describe explicitly a linear transformation from R3 into R3 which has as its
range the subspace spanned by (1, 0, -1) and (1, 2, 2).
Relevant Equations:: linear transformation

"There is a linear transformation T from R3 to R3 such that T (1, 0, 0) = (1,0,−1), T(0,1,0) = (1,0,−1) and T(0,0,1) = (1,2,2)" - why is this the case?

Thank you.

A linear transformation can be fully described by its action on any basis. Can you see why?

 

[jolly_math]

$$
\begin{bmatrix}
1 & 1 & 1 \\
0 & 0 & 2 \\
-1 & -1 & 2\\
\end{bmatrix}
\times
\begin{bmatrix}
x \\
y \\
z\\
\end{bmatrix}
$$

For R3, would
$$
\begin{bmatrix}
1 & 1 & 1 \\
0 & 2 & 0 \\
-1 & 2 & -1\\
\end{bmatrix}
\times
\begin{bmatrix}
x \\
y \\
z\\
\end{bmatrix}
$$
also work (switching which vector corresponds to each basis)? Thanks.

 

[Hall]

For R3, would
$$
\begin{bmatrix}
1 & 1 & 1 \\
0 & 2 & 0 \\
-1 & 2 & -1\\
\end{bmatrix}
\times
\begin{bmatrix}
x \\
y \\
z\\
\end{bmatrix}
$$
also work (switching which vector corresponds to each basis)? Thanks.

Yes, I think. I think even double columns of $(1,2,2)$ will also satisfy the given things.